# Definition:Special Linear Group

## Definition

Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$.

The special linear group of order $n$ on $R$ is the set of square matrices of order $n$ whose determinant is $1$.

It is a group under (conventional) matrix multiplication.

It is denoted $\SL {n, R}$, or $\SL n$ if the ring is implicit.

The ring itself is usually a standard number field, but can be any commutative ring with unity.

## Also known as

Some authors prefer $\operatorname{SL}_n \paren R$ and $\operatorname{SL}_n$ over $\SL {n, R}$.

## Also see

• Results about the special linear group can be found here.